关于丢番图方程x^4+mx^2y^2+ny^4=z^2(Ⅰ)

On the Diophantine Equation x^4+mx^2y^2+ny^4=z^2(Ⅰ)

利用Fermat无穷递降法 ,证明了方程x4+mx2 y2 +ny4=z2 在 (m ,n) =(6 ,± 12 ) ,(6 ,30 ) ,(- 12 ,- 12 ) ,(- 12 ,± 84)时均无正整数解 ,并且获得了方程在 (- 6 ,12 ) ,(± 12 ,12 ) ,(12 ,- 12 ) ,(± 6 ,6 )时的无穷多组正整数解的通解公式 ,从而完善了Aubry等人的结果 .

We get somenecessary conditions if the diophantine equations x4+mx2y2+ny4=z2 has positive Integer solutions that fit (x,y)=1 by using elementary theory of number and Fermat method of infinite descent.